Definition:Limit Point/Topology/Set/Definition 1

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Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.


A point $x \in S$ is a limit point of $A$ if and only if every open neighborhood $U$ of $x$ satisfies:

$A \cap \paren {U \setminus \set x} \ne \O$

That is, if and only if every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.


More symbolically, a point $x \in S$ is a limit point of $A$ if and only if

$\forall U\in \tau :x\in U \implies A \cap \paren {U \setminus \set x} \ne \O\text{.}$


Also see


Sources